IA Scholar Query: Identity Testing for constant-width, and commutative, read-once oblivious ABPs.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 15 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time
https://scholar.archive.org/work/gop5fmvbr5cunj6pdsjpiw62oi
Hrubeš and Wigderson [Hrubeš and Wigderson, 2015] initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, there are deterministic polynomial-time algorithms due to Garg, Gurvits, Oliveira, and Wigderson [Ankit Garg et al., 2016] and Ivanyos, Qiao, and Subrahmanyam [Ivanyos et al., 2018]. A central open problem in this area is to design an efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including concepts from matrix coefficient realization theory [Volčič, 2018] and properties of cyclic division algebras [T.Y. Lam, 2001]. En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas [Michael A. Forbes and Amir Shpilka, 2013] inside a cyclic division algebra of small index.V. Arvind, Abhranil Chatterjee, Partha Mukhopadhyay, Amit Chakrabarti, Chaitanya Swamywork_gop5fmvbr5cunj6pdsjpiw62oiThu, 15 Sep 2022 00:00:00 GMTOn Finer Separations Between Subclasses of Read-Once Oblivious ABPs
https://scholar.archive.org/work/veb2z2jevrak3gfejkyakg6b2e
Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials). Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank. Our proof of the above result builds on the results of Marinari, Möller and Mora (1993), and Möller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.C. Ramya, Anamay Tengse, Petra Berenbrink, Benjamin Monmegework_veb2z2jevrak3gfejkyakg6b2eWed, 09 Mar 2022 00:00:00 GMTBlack-box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial-time
https://scholar.archive.org/work/kexo434odjeejo5zs6tnyhg4ze
Hrubeš and Wigderson (2015) initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, deterministic polynomial-time algorithms are known for this problem following the works of Garg, Gurvits, Oliveira, and Wigderson (2016) and Ivanyos, Qiao, and Subrahmanyam (2018). A central open problem in this area is to design efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this problem for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including key concepts from matrix coefficient realization theory (Volčič, 2018) and properties of cyclic division algebra (Lam, 2001). En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas (2013) inside a cyclic division algebra of small index.V. Arvind and Abhranil Chatterjee and Partha Mukhopadhyaywork_kexo434odjeejo5zs6tnyhg4zeFri, 11 Feb 2022 00:00:00 GMTOn Finer Separations between Subclasses of Read-once Oblivious ABPs
https://scholar.archive.org/work/jzs7rblf3zdnbgm52ie6ngq2py
Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials). Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank. Our proof of the above result builds on the results of Marinari, Möller and Mora (1993), and Möller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.C. Ramya, Anamay Tengsework_jzs7rblf3zdnbgm52ie6ngq2pySat, 01 Jan 2022 00:00:00 GMT